Transcript Lesson 22 (Transformations)

Shifting
f ( x)  x
x
0
1 or -1
2 or -2
3 or -3
4 or -4
2
f(x)
0
1
4
9
16
f1 ( x)  x  3
2
x
0
1 or -1
2 or -2
3 or -3
4 or -4
f(x) + 3
3
4
7
12
19
(4, 19)
(4, 16)
f ( x)  x
2
f1 ( x)  x  3
2
Vertical Shifting
• If c is positive, then a graph of y = f(x) + c
will be shifted c units vertically upward.
• If c is positive, then a graph of y = f(x) – c
will be shifted c units vertically downward.
f ( x)  x
x
0
1 or -1
2 or -2
3 or -3
4 or -4
f 2 ( x)  ( x  3)
2
f(x)
0
1
4
9
16
x
-3
-4 or -2
-5 or -1
-6 or 0
- 7 or 1
2
f(x)
0
1
4
9
16
(4, 16)
f ( x)  x
2
(1, 16)
f 2 ( x)  ( x  3)
2
f ( x)  x
x
0
1 or -1
2 or -2
3 or -3
4 or -4
2
f(x)
0
1
4
9
16
f3 ( x)  ( x  3)
x
3
2 or 4
1 or 5
0 or 6
-1 or 7
2
f(x)
0
1
4
9
16
(4, 16)
f ( x)  x
2
(7, 16)
f3 ( x)  ( x  3)
2
Horizontal Shifting
• If c > 0, the graph of y = f(x – c) will be
shifted horizontally c units right from the
graph of y = f(x).
• If c > 0, the graph of y = f(x + c) will be
shifted horizontally c units left from the
graph of y = f(x).
REFLECTIONS
f ( x)  x
x
0
1 or -1
2 or -2
3 or -3
4 or -4
2
f(x)
0
1
4
9
16
f 4 ( x)   x
x
0
1 or -1
2 or -2
3 or -3
4 or -4
2
f(x)
0
-1
-4
-9
-16
f 4 ( x)   x
2
(4, 16)
(4, -16)
f ( x)  x
2
Vertical Reflection
• The graph of y = -f(x) will be reflected
vertically over the x-axis from y = f(x).
g ( x)  x
x
f(x)
0
0
1
1
4
2
9
3
16
4
g1 ( x)   x
x
f(x)
0
0
-1
1
-4
2
-9
3
-16
4
(4, 2)
g ( x)  x
(-4, 2)
g1 ( x)   x
Horizontal Reflection
• The graph of y = f(-x) would be reflected
horizontally about the y-axis from y = f(x).
STRETCHING/COMPRESSING
f ( x)  x
x
0
1 or -1
2 or -2
3 or -3
4 or -4
2
f(x)
0
1
4
9
16
f5 ( x)  2 x
x
0
1 or -1
2 or -2
3 or -3
4 or -4
2
f(x)
0
2
8
18
32
(2, 4)
f ( x)  x
2
(2, 8)
f5 ( x)  2 x
2
1 2
f 6 ( x)  x
2
x
f(x)
0
0
1 or -1
½
2 or -2
2
3 or -3
9/2
4 or -4
8
(4, 16)
(4, 8)
f ( x)  x
2
1 2
f 6 ( x)  x
2
Vertical Stretch or Compression
• If c > 1, the graph of y = c f(x) will be
stretched vertically by a factor of c from
the graph of y = f(x).
• If 0 < c < 1, the graph of y = c f(x) will be
compressed vertically by a factor of c from
the graph of y = f(x).
x
f(x)
0
0
1
1
4
2
9
3
16
4
g ( x)  x
x
0
½
2
9/2
8
f(x)
0
1
2
3
4
g2 ( x)  2 x
(5, 10)
(10, 10)
(4, 2)
g ( x)  x
(2, 2)
g2 ( x)  2 x
x
f(x)
0
0
1
1
4
2
9
3
16
4
g ( x)  x
x
0
2
8
18
32
f(x)
0
1
2
3
4
1
g3 ( x) 
x
2
(10, 10)
(4, 2)
g ( x)  x
(8, 2)
1
g3 ( x) 
x
2
Horizontal Stretch or Compression
• If c > 1, the graph of y = f(cx) will be
horizontally compressed by a factor of 1/c
from the graph of f(x).
• If 0 < 1/c < 1, the graph of y = f((1/c)x) will
be horizontally stretched by a factor of c
from the graph of f(x).